Structure of three-loop contributions to the beta-function of N=1 SQED with N_f flavors, regularized by the dimensional reduction

TL;DR

This paper compares three-loop beta-function calculations in N=1 SQED with N_f flavors using higher derivative regularization and dimensional reduction, revealing scheme-dependent structures and confirming known results.

Contribution

It demonstrates how three-loop contributions to the beta-function can be derived in dimensional reduction and compares them with higher derivative regularization results.

Findings

Three-loop beta-function contains scheme-dependent terms proportional to (N_f)^2.

The known three-loop beta-function is recovered after including scheme-independent terms.

Higher derivative regularization simplifies the derivation of the NSVZ relation.

Abstract

In the case of using the higher derivative regularization for $N=1$ SQED with $N_f$ flavors the loop integrals giving the $\beta$-function are integrals of double total derivatives in the momentum space. This feature allows to reduce one of the loop integrals to an integral of the $\delta$-function and to derive the NSVZ relation for the renormalization group functions defined in terms of the bare coupling constant. In this paper we consider $N=1$ SQED with $N_f$ flavors regularized by the dimensional reduction in the $\overline{\mbox{DR}}$-scheme. Evaluating the scheme-dependent three-loop contribution to the $\beta$-function proportional to $(N_f)^2$ we find the structures analogous to integrals of the $\delta$-singularities. After adding the scheme-independent terms proportional to $(N_f)^1$ we obtain the known result for the three-loop $\beta$-function.